贝尔法斯特圣玛丽学院国际商务文凭证书【购证 微kaa77788】pi/2

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11: 10.39 Relations to Other Functions

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$I_{\frac{1}{2}}\left(z\right)={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\sinh z,$

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$I_{-\frac{1}{2}}\left(z\right)={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\cosh z,$

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10.39.2 $$K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/10.39.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.4 $$K_{\frac{3}{4}}\left(z\right)=\frac{1}{2}{\pi }^{\frac{1}{2}}{z}^{-\frac{3}{4}}\left(\frac{1}{2}U(1,2z^{\frac{1}{2}})+U(-1,2z^{\frac{1}{2}})\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.39.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.8 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(2z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.48

Referenced by:

(9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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12: 4.23 Inverse Trigonometric Functions

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4.23.27 $$\arctan \left(\mathrm{i}y\right)=\pm \frac{1}{2}\pi +\frac{\mathrm{i}}{2}\ln \left(\frac{y+1}{y-1}\right),$$ $y\in (-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\cup$: union and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.23.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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Table 4.23.1: Inverse trigonometric functions: principal values at 0, $\pm 1$, $\pm \mathrm{\infty }$.

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$x$	$\arcsin x$	$\arccos x$	$\arctan x$	$\mathrm{arccsc}x$	$\mathrm{arcsec}x$	$\mathrm{arccot}x$
$-\mathrm{\infty }$	–	–	$-\frac{1}{2}\pi$	$0$	$\frac{1}{2}\pi$	$0$
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$0$	$0$	$\frac{1}{2}\pi$	$0$	–	–	$\mp \frac{1}{2}\pi$
$1$	$\frac{1}{2}\pi$	$0$	$\frac{1}{4}\pi$	$\frac{1}{2}\pi$	$0$	$\frac{1}{4}\pi$
$\mathrm{\infty }$	–	–	$\frac{1}{2}\pi$	$0$	$\frac{1}{2}\pi$	$0$

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13: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.4 $$\arctan z=\pm \frac{\pi }{2}-\frac{1}{z}+\frac{1}{3z^3}-\frac{1}{5z^5}+\mathrm{\cdots },$$ $\mathrm{\Re }z\gtrless 0$, $|z|\ge 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error. $\frac{\pi }{2}$ should have a negative sign when .)

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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14: 4.40 Integrals

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4.40.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}\frac{\sin \left(ax\right)}{\sinh x}dx=\frac{1}{2}\pi \coth \left(\frac{1}{2}\pi a\right)-\frac{1}{a},$$ $a\ne 0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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15: 20.10 Integrals

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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16: 12.7 Relations to Other Functions

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12.7.5 $$U(\frac{1}{2},z)=D_{-1}\left(z\right)=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{\frac{1}{4}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right),$$

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Referenced by:

§12.11(ii)

Permalink:

http://dlmf.nist.gov/12.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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12.7.6 $$U(n+\frac{1}{2},z)=D_{-n-1}\left(z\right)=\sqrt{\frac{\pi }{2}}\frac{{(-1)}^n}{n!}{\mathrm{e}}^{-\frac{1}{4}{z}^2}\frac{d^n\left({\mathrm{e}}^{\frac{1}{2}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right)\right)}{{dz}^n},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/12.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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12.7.10 $$U(0,z)=\sqrt{\frac{z}{2\pi }}{K}_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.9 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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17: 7.7 Integral Representations

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7.7.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{2a}{\mathrm{e}}^{-2ab},$$ , .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.3 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.13 $$\mathrm{f}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{3}{4}\right)\mathrm{\Gamma }\left(\frac{1}{4}-s\right)ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.14 $$\mathrm{g}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{1}{4}\right)\mathrm{\Gamma }\left(\frac{3}{4}-s\right)ds.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\cos \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.22 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\sin \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Keywords:

Laplace transform

A&S Ref:

7.4.23 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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18: 18.16 Zeros

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18.16.2 $${\theta }_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\frac{1}{2})\pi }{n+\frac{1}{2}}\le {\theta }_{n,m}^{(\alpha ,\beta )}\le \frac{m\pi }{n+\frac{1}{2}}={\theta }_{n,m}^{(\frac{1}{2},-\frac{1}{2})},$$ $\alpha ,\beta \in [-\frac{1}{2},\frac{1}{2}]$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and ${\theta }_{n,m}$: zeros

Referenced by:

§18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)

Permalink:

http://dlmf.nist.gov/18.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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18.16.3 $${\theta }_{n,m}^{(-\frac{1}{2},-\frac{1}{2})}=\frac{(m-\frac{1}{2})\pi }{n}\le {\theta }_{n,m}^{(\alpha ,\alpha )}\le \frac{m\pi }{n+1}={\theta }_{n,m}^{(\frac{1}{2},\frac{1}{2})},$$ $\alpha \in [-\frac{1}{2},\frac{1}{2}]$, $m=1,2,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and ${\theta }_{n,m}$: zeros

Referenced by:

§18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)

Permalink:

http://dlmf.nist.gov/18.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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18.16.4 $\alpha ,\beta \in [-\frac{1}{2},\frac{1}{2}]$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ and ${\theta }_{n,m}$: zeros

Permalink:

http://dlmf.nist.gov/18.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

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19: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.3 $${𝐄}_{\nu }\left(z\right)\sim -Y_{\nu }\left(z\right)-\frac{1+\cos \left(\pi \nu \right)}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu (1-\cos \left(\pi \nu \right))}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}},$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Proof sketch:

Derivable from (11.10.16).

Permalink:

http://dlmf.nist.gov/11.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

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11.11.11 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\mathrm{e}}^{-\nu \mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_k{b}_k(\lambda )}{{\nu }^k},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Sources:

Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352)

Proof sketch:

Derivable by applying Laplace’s method to the integral (11.10.4).

Permalink:

http://dlmf.nist.gov/11.11.E11

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.15 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\left(\frac{1+\sqrt{1-{\lambda }^2}}{\lambda }\right)}^{\nu }\frac{{\mathrm{e}}^{-\nu \sqrt{1-{\lambda }^2}}}{{(1-{\lambda }^2)}^{1/4}},$$ , $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Source:

Olver (1997b, pp. 103 and 352)

Permalink:

http://dlmf.nist.gov/11.11.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint has been extended to include $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-06 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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20: 5.7 Series Expansions

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5.7.5 $$\psi \left(1+z\right)=\frac{1}{2z}-\frac{\pi }{2}\cot \left(\pi z\right)+\frac{1}{z^2-1}+1-\gamma -\sum _{k=1}^{\mathrm{\infty }}(\zeta \left(2k+1\right)-1)z^{2k},$$ , $z\ne 0,\pm 1$.

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.15

Referenced by:

§5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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